# differentiation rules

Here follow the rules for differentiating common functions of a single real variable that every student of calculus should know. In each case the variable with respect to which the derivative is being taken is $$x$$, both $$u$$ and $$v$$ are functions of $$x$$,and $$c$$ is a constant. Note that $$u^{\prime}$$ should be understood as $$\displaystyle\frac{du}{dx}$$.

$\begin{eqnarray*} \frac{d}{dx}\left( c \right) & = & 0 \\ & & \\ \frac{d}{dx}\left( x \right) & = & 1 \\ & & \\ \frac{d}{dx}\left( cu \right) & = & cu^{\prime} \\ & & \\ \frac{d}{dx}\left( u \pm v \right) & = & u^{\prime} + v^{\prime} \\ & & \\ \frac{d}{dx}\left( uv \right) & = & u^{\prime}v + uv^{\prime} \\ & & \\ \frac{d}{dx}\left( \frac{u}{v} \right) & = & \frac{u^{\prime}v - uv^{\prime}}{v^2} \\ & & \\ \frac{d}{dx}\left( u^n \right) & = & nu^{n-1}u^{\prime} \\ & & \\ \frac{d}{dx}\left( \ln u \right) & = & \frac{u^{\prime}}{u} \\ & & \\ \frac{d}{dx}\left( \sin u \right) & = & (\sin u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \cos u \right) & = & (-\sin u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \tan u \right) & = & \left(\sec^2 u \right) u^{\prime} \\ & & \\ \frac{d}{dx}\left( \cot u \right) & = & -\left( \csc^2 u\right) u^{\prime} \\ & & \\ \frac{d}{dx}\left( \sec u \right) & = & (\sec u \tan u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \csc u \right) & = & -(\csc u \cot u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \arcsin u \right) & = & \frac{u^{\prime}}{\sqrt{1-u^2}} \\ & & \\ \frac{d}{dx}\left( \arccos u \right) & = & \frac{-u^{\prime}}{\sqrt{1-u^2}} \\ & & \\ \frac{d}{dx}\left( \arctan u \right) & = & \frac{u^{\prime}}{1+u^2} \\ & & \\ \frac{d}{dx}\left( \mbox{arccot } u \right) & = & \frac{-u^{\prime}}{1+u^2} \\ & & \\ \frac{d}{dx}\left( \mbox{arcsec } u \right) & = & \frac{u^{\prime}}{|u|\sqrt{u^2-1}} \\ & & \\ \frac{d}{dx}\left( \mbox{arccsc } u \right) & = & \frac{-u^{\prime}}{|u|\sqrt{u^2-1}} \\ & & \\ \end{eqnarray*}$